Hi, and welcome. My name is Anthony Varela. And today we are going to be writing a system of linear equations. So we're going to look at a situation and write a couple of linear equations representing that scenario. We will be defining variables within the context we are given, and we're going to determine, based on our variable definitions, if our equations could be considered together in a system.
So our situation involves a carnival for the whole family. And the organization that puts together this carnival has two different price levels-- $3 tickets for kids and $5 tickets for adults. And this is what they know about to their event. The total attendance was 234 people. And the total ticket sales was $850. So a combination of kids and adults came, so that total attendance was 234 people. And a combination of $3 tickets and $5 tickets were sold to add up to $850 in ticket sales.
So let's go ahead and define some variables that we're going to use in our equations. Now what do we not know? We don't know exactly how many kids came, and we don't know exactly how many adults came. We just know those totals. So we're going to define our variables as the number of kids and the number of adults in attendance. And let's go ahead and use x and y. So x equals number of kids in attendance, y equals number of adults in attendance. So right off the bat, I know that x plus y must equal 234. Number of kids plus number of adults has to add up to 234 people.
So that's one of the equations. Let's go ahead and write an equation to represent the ticket sales. Well we can multiply the cost of the ticket by our variables, x and y, to represent cost-- or the revenue in ticket sales. So 3x is going to be ticket sales from kids, and 5y is going to be in ticket sales from adults, just multiplying the cost of that ticket by how many kids or adults attend. And we're going to add those two figures together to get $850.
So we would say, then, that 3x plus 5y equals 850. So our first equation here relates the ticket sales broken down into ticket sales from kids plus ticket sales from adults equals total ticket sales. And our second equation relates the attendance-- number of kids plus number of adults equals the total number of people that were there, 234. So this is my system of equations. And I know that it's a system because my corresponding variables represent the same quantity or unit. In both equations, x represents number of kids, and y represents number of adults.
Our next situation has to do with buying T-shirts. So an organization is interested in buying medium and large T-shirts, and they're pricing out their options. And from one vendor, medium T-shirts cost $12, and large T-shirts cost $16. Now I don't know exactly how many of each they're buying, but I do know that the total price tag is $836 for a combination of medium and large T-shirts.
Now they looked at a second vendor, and the prices were different. Medium t-shirts were $13, and large T-shirts were $15. The total price, then, for this combination of medium and large T-shirts was $824. So we can write a system of equations that relates these two quantities and prices.
So we're going to define our variables. We'll say, then, that x equals the number of medium T-shirts, and y equals the number of large T-shirts. Those are the quantities that I don't know. I don't know how much of each size the company is buying.
So I'm going to say, then, that 12x is going to represent the cost of medium T-shirts, multiplying that dollar amount by the number of medium T-shirts. I'm going to add to that 16y, getting $16 for every large T-shirt. And adding those two costs together is the total price at the first vendor, $836. So this equation represents the cost of T-shirts at vendor A.
How about the vendor B, the second option? Well we can say, then, that 13x in this case represents the cost of medium T-shirts because $13 is the price of one medium T-shirt. X is the number of medium T-shirts. So then we'll add to that, 15y-- the price of a large T-shirt and the quantity of large T-shirts. And these two individual costs have to come together for a total price of $824. Once again, this is a system of equations because in both equations, x represents number of medium T-shirts and y represents number of large T-shirts. So we can consider these two equations together at the same time.
So let's review writing a system of linear equations. Well step one, we want to define all variables from the situation. What do we not know? Assign a variable to it and give it a very clear definition. Step two, we would then construct equations to represent our scenario. So thinking about cost and quantity, and you can multiply them together to get total cost.
Step three, look at what the variables mean in each equation. And remember, if our corresponding variables represent the same quantities or units, that means they have matching definitions. The equations form a system. They can be considered together at the same time. So thanks for watching this tutorial on writing a system of linear equations. Hope to see you next time.